A Recurrence Matrix Solution for the Dynamic Response of Elastic Aircraft

Abstract

A systematic procedure is developed for the calculation of the structural response of an airplane subject to dynamic loads. Particular attention is given the problem of determining the stresses developed due to flight through gusts. Difference equivalents for derivatives and matrix notation are used to develop a recurrence relation that permits step-by-step calculation of the response and of the loads that occur on the structure. The chief feature of this recurrence approach is that the generality and physical aspects of the basic equilibrium relations of the problem are preserved without loss of ease in application. The use of difference equivalents for derivatives in the solution of dynamic problems is first illustrated by means of a simple damped oscillator example, and the application to the flexible aircraft structure is then made. For brevity, the case of wing bending and vertical motion of the airplane is treated, although the method may be readily extended to take into account also wing torsional deformations, pitching motion of the airplane, fuselage deflections, and tail forces of known character. Either a sharp-edge gust or a gust of arbitrary shape in the spanwise or flight directions may be treated. Some results obtained by application of the recurrence matrix relation are presented, and the advantages of this method over other methods of evaluating the dynamic response of an aircraft are discussed.